The government has insisted it will not intervene in the female bishops controversy by using equalities legislation to force the Church of England to change its internal rules.

The prime minister's spokesman said David Cameron shared the disappointment of many at the vote on female bishops, but said the issue was a matter for the church to decide.

Sources close to the culture secretary, Maria Miller, who is also equalities minister, issued strongly worded criticism after the vote on Tuesday night to open up the episcopate to female clerics was lost by just six votes.

But it was stressed by No 10 that this displeasure would not extend to direct intervention in the internal workings of the Church of England.

Allies of Miller said: "Whilst this is a matter for the church, it is very disappointing … As we seek to help women fulfil their potential throughout society this ruling would suggest the church is at very least behind the times."

Philip Hammond, the defence secretary, said the Church of England must resolve the issue of female bishops: "Otherwise it's going to increasingly find itself marginalised."

The Labour MP and active Christian Chris Bryant has suggested he will introduce legislation that would require the CofE to be compliant with UK equality laws.

He said he feared the church may have died as a national institution as a result of the vote.

Another former minister, Ben Bradshaw, claimed the church was "being held hostage" by evangelicals and conservatives. "This means the church is being held hostage by an unholy and unrepresentative alliance of conservative evangelicals and conservative Catholics," Bradshaw said.

The church has an exemption from equalities and employment legislation allowing it to disbar women from the episcopate. If that immunity was removed, it would be possible for women to sue the CofE for discrimination.

It would not be possible for such legislation to be pushed through parliament without the support of the government, although MPs such as Bryant could introduce 10-minute rule bills, or a private member's bill.

Senior Liberal Democrats, including the chief whip, Alistair Carmichael, questioned whether bishops should be allowed to stay in the House of Lords if they were going to remain male-only appointments.